In Nodes, edges, graphs & rules: the basic concepts of Wolfram Physics I introduced a simple rule.
When I applied that rule to a simple universe, it quickly evolved into a surprisingly complex universe.
But why that particular rule?
Why not some other rule?
What other rules could be applied to a universe?
And what other universes would arise from these other rules?
Today, I want to explore different rules, different universes.
Billiard balls
Let’s start by being a bit more concise about that original rule.
Remember when I described the rule in words?
Something like:
“find two edges that come from the same node, delete one of the two edges, create a new node, and create a new edge from each of the three existing nodes to the new node”
That’s quite a mouthful.
Let’s run through the rule again, but this time let’s number the nodes, so that it’s clearer which ones I’m talking about.
Here are the three nodes and two edges we started with:
When we find edges that match the rule, we show those edges in blue:
Let’s number the nodes ➊, ➋ and ➌:
Now it’s easier to describe the matched edges. One is from node ➊ to node ➋. The other is from node ➊ to node ➌.
I can describe the matched edges in symbols:
➊➝➋ ➊➝➌
These billiard balls and arrows are just a shorter way of saying:
“find two edges that come from the same node”
So now when we delete one edge, shown in red, and keep the other, shown in blue, we can say that we’re deleting edge ➊➝➋ and keeping edge ➊➝➌:
And when we create a new node and three new edges, shown in green, we can number the new node ➍ say that we’re creating edges ➊➝➍, ➋➝➍ and ➌➝➍:
I can describe the entire rule in symbols:
➊➝➋ ➊➝➌ → ➊➝➌ ➊➝➍ ➋➝➍ ➌➝➍
The billiard balls after the arrow, ➊➝➌ ➊➝➍ ➋➝➍ ➌➝➍, are just a shorter way of saying:
“delete one of the two edges, create a new node, and create a new edge from each of the three existing nodes to the new node”
For those of you who prefer words to numbers, the billiard balls might not seem much of an improvement over my original description.
But they’re going to help us define rules more concisely.
And they’re going to help us run through all possible rules, to find the ones that are the most interesting.
The most boring rule
As it turns out, some rules are mind-numbingly boring.
Let’s start with the simplest rule there is:
➊➝➋ → ➊➝➋
What this rule means is:
“find an edge and leave it exactly as it is”
Let’s apply this rule to a simple universe:
First, we need to find an edge. Oh, look, there’s one!
So we have our edge ➊➝➋. Now let’s leave it exactly as it is.
If we apply this mind-numbingly boring rule over and over again, precisely nothing happens.
I’m glad I don’t live in that universe.
Ever-so-slightly less boring rules
Let’s try another rule:
➊➝➋ → ➋➝➊
What this rule means is:
“find an edge, delete it, and create a new edge between the same two nodes, but pointing the opposite way”
All this rule is going to do to our simple universe is flip-flop the edge so that it’s pointing one way, then the other way, then back the original way, and so on, ad infinitum:
Here’s another rule:
➊➝➋ → ➊➝➌
which means:
“find an edge, delete it, create a new node, and create a new edge from the first node of the matched edge to the new node”
You can see what’s going to happen here. The rule will continually delete the existing edge and create a new edge:
We’re not getting anywhere here, and it’s not difficult to see why.
Any rule that starts out with one edge and ends up with one edge is going to leave the universe with the same number of edges as before.
These are dead-end rules. They give dead-end universes.
But you can see where we’re going with the billiard balls.
The numerical notation allows us to try out every possible rule, from dead-end rules like these on up.
Which allows us to take a look at every possible universe, from dead-end universes like these on up.
Some of these universes will be more interesting than others.
Pretty rules
Let’s try a rule that starts out with one edge and ends up with two:
➊➝➋ → ➊➝➋ ➊➝➌
Unlike our dead-end rules, this one is clearly going to give a universe that expands:
“find an edge, create a new node, and create a new edge from the first node of the matched edge to the new node”
Starting with one edge, we create a second edge, then a third edge, then a fourth edge, all coming from the same node:
We can do this all day:
I think of this universe as a starburst universe.
Here’s a rule that does the same thing the other way around:
➊➝➋ → ➊➝➋ ➌➝➋
It’s almost the same rule, but with a subtle reversal:
“find an edge, create a new node, and create a new edge from the new node to the second node of the matched edge”
Starting with one edge, we create a second edge, then a third edge, then a fourth edge, all going to the same node:
This gives a reverse starburst universe:
Lonesome rules
Some rules give disconnected universes:
➊➝➋ → ➊➝➋ ➌➝➍
This rule finds an edge:
Then it creates two new nodes and creates a new edge between those two new nodes:
Which leaves us with two disconnected edges:
In a way, these are two separate universes. There’s no connection between them, which means that there’s no communication between them, which makes it meaningless to think of them as the same universe. Unless, of course, another rule operates on the graph to stitch these universes back together, or to create tenuous bridges between otherwise disconnected universes.
If we carry on applying this rule, it gives an ever larger number of separate universes:
In the case of this rule, each of these universes is an extremely boring universe, with just two nodes and one edge. But you can imagine a rule that gives multiple universes, each of which is an extremely interesting universe, with as many nodes and edges as our own universe.
Arboreal rules
This rule gives a branching, tree-like universe:
➊➝➋ → ➊➝➋ ➋➝➌
So does this rule, except that the branching, tree-like universe is different, reminiscent of sparklers:
➊➝➋ → ➋➝➊ ➌➝➋
Straight-down-the-line rules
Here’s another rule. This one gives a straight-line universe:
➊➝➋ → ➊➝➌ ➌➝➋
If, instead of starting with a single edge, you start with three edges forming a triangle, the same rule gives a polygon, which eventually smooths out into a circle:
So you see that the same rule applied to a different initial graph can give different universe.
Which raises a couple of questions.
What should we choose as the initial graph?
And how might our choice change the universe we end up with?
One rule to rule them all
You see how this goes.
We’ve tried out some very simple rules here.
And we’ve ended up with some very different universes.
Some have been dead-end universes.
Some have been pretty, expansionary universes.
But none of the universes we’ve explored today looks very much like our own universe.
If there’s one rule for our universe, none of the rules we’ve explored today is it.
What does our universe look like?
As ever with our explorations of Wolfram Physics, this raises more questions.
If there’s one rule for our universe, what is that rule?
Or are there multiple rules for our universe?
Maybe many rules are involved in the evolution of our universe.
Maybe all rules are involved in the evolution of our universe.
And here’s another question.
How will we know when we have the answers to those questions?
How will we know when we’ve correctly identified the one rule, or the many rules, or all the rules that apply to our universe?
And here are yet more questions.
How can I be so confident that none of the universes we’ve explored today looks very much like our own universe?
What would convince me that these nodes, edges, graphs and rules do look like our own universe?
At first glance, even the more complex graph we started out with bears little resemblance to our universe. What do nodes and edges have to do with the world we live in: the world of butterflies and bulldozers and dishcloths and dinosaurs? (All right, there might not be dinosaurs any more, but there are definitely dishcloths.)
To answer these questions, we have to ask an even more basic question.
What exactly does our universe look like?